The coordinates of the critical points of spin-S Ising models with coupling constants J and J′ are calculated for 1/2 ≤ S ≤ 13/2. The calculations are performed for several values of S and Δ ≡ J′/J independently by using the phenomenological renormalization-group method or (approximate) self-duality. Numerical results combined with a mean-field analysis show that the critical coupling strength for Δ ∼ 1 (weakly anisotropic lattice) is Kc(S) (Δ) = Kc(S) (1)[1 + a(1 − Δ)], where a = (d − 1)/d is independent of S (d is the space dimension). Both free energy and internal energy are determined at the critical points. An extremum of the critical internal energy is found at Δ* ∈ (0, 1). The parameter Δ* can be used as a criterion that separates quasi-isotropic and quasi-one-dimensional regimes (Δ* < Δ ≤ 1 and Δ < Δ*, respectively). The finite-size scaling amplitudes As and Ae of the inverse spin-spin and energy-energy correlation lengths are estimated. Calculations show that the amplitudes As and Ae are independent of S within the accuracy of the adopted approximations. Moreover, their ratio Ae/As is independent of the anisotropy parameter Δ. These results support the Ising universality hypothesis.